Accessories such as bags or organizers are often used in connection with portable containers to provide additional functionality to the container. In particular, such accessories are frequently used in connection with plastic buckets, e.g., the ubiquitous five-gallon bucket, but may be used in connection with other containers. Additionally, such accessories frequently contain multiple storage compartments to provide storage for tools, parts, supplies, or other articles that are used in conjunction with the bucket. Such accessories are typically mounted on the bucket by either draping the accessory over the rim of the bucket or constricting the accessory around the outer sidewall of the bucket. Both of these mounting techniques suffer from drawbacks. Accessories that are draped over the top of the bucket prevent the use of a lid in connection with the bucket and may fall off of the bucket if the bucket is inverted or partially inverted, among other drawbacks. Accessories that are constricted around the outer sidewall of the bucket can tend to slip, among other drawbacks. Further, there is a lack of articles that can provide for directly mounting accessories other than the aforementioned bags and organizers on a plastic bucket or other container. The articles and methods described herein can address these and other problems with existing container accessories.